arXiv:hep-ph/9802319v1 13 Feb 1998
Derivative expansions of real-time thermal e?ective actions
Maria Asprouli, Victor Galan-Gonzalez Imperial College Theoretical Physics Group Blackett Laboratory London SW7 2BZ February 1, 2008
Abstract In this work we use a generalised real-time path formalism with properly regularised propagators based on Le Bellac and Mabilat  and calculate the e?ective potential and the higher order derivative terms of the e?ective action in the case of real scalar ?elds at ?nite temperature. We consider time-dependent ?elds in thermal equilibrium and concentrate on the quadratic part of the expanded e?ective action which has been associated with problems of non-analyticity at the zero limits of the four external momenta at ?nite temperature. We derive the e?ective potential and we explicitly show its independence of the initial time of the system when we include both paths of our time contour. We also derive the second derivative in the ?eld term and recover the Real Time (RTF) and the Imaginary Time Formalism (ITF) and show that the divergences associated with the former are cancelled as long as we set the regulators zero in the end. Using an alternative method we write the ?eld in its Taylor series form and we ?rst derive RTF and ITF in the appropriate limits, check the analyticity properties in each case and do the actual time derivative expansion of the ?eld up to second order in the end. We agree with our previous
results and discuss an interesting term which arises in this expansion. Finally we discuss the initial time-dependence of the quadratic part of the e?ective action before the expansion of the ?eld as well as of the individual terms after the expansion.
The interest in the amalgamation of ?eld theory and statistical mechanics arose from the realisation that many problems encountered experimentally and theoretically in particle physics have many-body aspects. For this reason, zero-temperature quantum ?eld theory was reformulated by generalising the usual time-ordered products of operators to the ordered products along a path in a complex time-plane . The choice of the path gives rise to di?erent formalisms but all theories should give the same physical answers. Although the various path-ordered ?nite temperature ?eld theory formalisms such as Real Time Formalism (RTF) including the closed-time approach  and Imaginary Time Formalism (ITF)  should give the same physics, there has been serious discussion about their exact equivalence. In this paper we will tackle a problem in RTF which consists in the occurrence of pathologies associated with singularities, arising in diagrams with self-energy insertions or in some e?ective potential calculations. This problem appears when products of delta functions with the same argument are involved and creates non-analyticities in the e?ective action at ?nite temperature, thus making it ill-de?ned. The main interest for developing an e?ective formalism which describes the ?nite temperature ?eld theory comes from the need to tackle important problems in phase transitions, which have played a crucial role in the early evolution of the universe. The signi?cance and observable quantities of a speci?c transition depend on its detailed nature and its order. A reason for a well-de?ned e?ective action comes from the fact that it represents the quantum corrections which in general might be of extreme importance in de?ning or changing the order of a transition. For example, analytic analysis  suggests that the electroweak phase transition is ?rst order because of quantum corrections from gauge bosons while non-perturbative lattice simulations of high temperature electroweak theory suggests that this is only true if the 2
Higgs is lighter than 70Gev . The inclusion of higher derivative terms in the derivative expansion of the e?ective action in a ?rst order transition is of great importance in cases such as the derivation of the rate of the sphaleron ?uctuations. These con?gurations have been used to explain the observed baryon asymmetry of the universe . Moreover, although the e?ective potential can give the approximate critical temperature of a given transition, it is not adequate for answering questions concerning the departure of the ?eld from equilibrium occurring during dynamical cooling near and below the critical temperature Tc . The e?ective potential describes static properties and therefore it is not an appropriate tool for studying the dynamical behaviour of a wide class of ?eld theory models considered in in?ationary scenarios. The standard method for estimating the quantum corrections is to ?rst integrate out quantum ?uctuations about a constant background. This gives an e?ective potential for φ which is then used in the equations of motion ?x determining φ(?, τ ) . Integrating out ?uctuations about a general inhomogeneous con?guration gives the full e?ective action which includes the higher derivative terms. In the language of quantum ?eld theory at ?nite temperature the e?ective potential Γ is given by Γ = βF , where F is the minimum of the free energy at which the system lies in the case of local thermal equilibrium. If the ensemble averages of the matter ?elds are homogeneous and static, then the free energy is given by the ?nite temperature e?ective potential . In analogy to quantum mechanics, the decay rate of an unstable con?guration φf with energy Ef is given by  2 1 h Γ = ? Im[ Ef ] = ?2Im[ lim ln φf | e?HT /? |φf ] T →∞ T h ? where H is the Hamiltonian and the matrix element can be described as a functional integral
h φf | e?HT /? |φf = N h Dφe?S[φ]/?
in Euclidean time. Here φ is subject to the condition φ(T /2) = φ(?T /2) = φf and S denotes the Euclidean action. Evaluating the functional integral
? to one loop, we expand S(φ) about a solution of the equations of motion, φ, ? we obtain and keeping only terms quadratic in the ?uctuations δφ = φ ? φ,
1 ? h h ? Dφe?S[φ]/? ? Ne?S(φ/? ) [ det(??? ?? + V ′′ (φ)) ]? 2
? h ≡ exp[ ?Sef f (φ/? ) ] where Sef f is the e?ective action. If we expand Sef f about a constant φ, i.e. in powers of momentum about a point with zero external momenta, in position space and zero temperature this reads Sef f (φ) = 1 d4 x[ ?Vef f (φ) + Z(φ)?? φ? ? φ + O((?? φ)4 ) ] 2
where we have made use of T = 0 Lorentz properties. For constant φ only the e?ective potential term survives. Although such an expansion up to the second derivative has been performed at zero temperature for scalar and Dirac ?eld theories , there are di?culties arising in the equivalent expansion at ?nite temperature. Das and Hott  ?nd a non-analyticity in the two-point functions involving the temperature dependent term of the quadratic part of the e?ective action. In this spirit, if the derivative expansion breaks down at ?nite temperature, the definition of the e?ective potential might not be unique. This non-analyticity, in the case of the vacuum polarisation for a scalar ?eld coupled to a classical external ?eld, manifests itself in the di?erence between the order of the limits (p0 → 0, p → 0) and (p → 0, p0 → 0) of the external momenta, the ?rst relating to the electric screening mass of the photon and the second to the plasma frequency of the particular ?eld theory under consideration . This non-commutativity appears in hot QCD , self interacting scalars [15, 16] and gauge theories with chiral fermions . In ITF, setting p? = 0 ?rst and performing the mode sum gives the same result as taking the limit p0 → 0 ?rst and the limit p → 0 afterwards. In RTF extra Feynman rules have been imposed to explain this di?erence in the two limits . The problem is neither due to subtleties in the use of Feynman parametrisation at ?nite temperature , nor to the in?nite number of possible extensions of p0 to the imaginary axis and its analytic continuation to the complex plane . The 4
lack of analyticity and the infrared divergences occurring in the de?nition of the e?ective action at ?nite temperature show the need for an e?ective ?eld theory formalism from which RTF and ITF rules can be derived easily. In the next chapter we will describe a method dealing with these problems. We will calculate the two-propagator contribution (bubble diagram) to the second derivative term in the e?ective action using Le Bellac and Mabilat’s generalised real-time path formulation with properly regularised propagators .
In Le Bellac and Mabilat’s approach  they derive Feynman rules that take explicitly into account the vertical part of the contour and recover the RTF in the case of diagrams with at least one ?nite external line and the ITF in the case of vacuum ?uctuations. They keep the regulators of the propagators when they ?nd problematic products of delta functions with the same argument and show that they can use RTF when no such problems arise. They claim that the contribution of the vertical part of the contour lies in the cancellation of the ti dependent terms of the horizontal part since the whole result should be ti and tf independent due to the KMS condition of the propagators. We will now describe in detail this method, which we will use throughout this paper.
Outline of the method
The speci?c approach uses the Mills  mixed representation of the propagators for a free scalar ?eld, with t de?ned in the generalised time path C, starting at ti and ending at ti ? iβ of Fig.1. The propagator is written as Dc (t, k) = dk0 ?ik0 t e [ θc (t) + n(k0 ) ]ρ(k0 , k) 2π (1)
where θc (t) is a contour θ function, n(k0 ) is the Bose-Einstein distribution function given by 1 n(k0 ) = βk0 (2) e ?1 5
?m(τ ) 6 ti CH
C ? V ti ? iβ
Figure 1: The integration contour C in the complex t-plane. and ρ(k0 , k) is the (temperature independent) two-point spectral function given by 2 2 ρ(k0 , k) = 2πε(k0 )δ(k0 ? ωk ) (3) where ε(k0 ) is the sign function and
2 ωk = k2 + m2
However, the propagators need to be regularised because eventually we want to take the Fourier transform in time by taking the limits ti → ?∞, tf → +∞ (to get energy conservation) and this is ill-de?ned since the integrands are linear combinations of complex exponentials. For this reason we write the δ distribution in its regularised form δ(k0 ? ωk ) = 1 1 1 [ ? ] 2iπ k0 ? ωk ? iε k0 ? ωk + iε (5)
and thus ρ(k0 , k) in Eq. (3) can be written as ρ(k0 , k) = i 2ωk rs r,s=±1 k0 ? sωk + iεr (6)
The regularised propagator can be written as
c > < DR (t, k) = DR (t, k)θc (t) + DR (t, k)θc (?t)
and obeys the KMS condition
> < DR (t ? iβ, k) = DR (t, k)
> Momentum integration in the complex k0 -plane will give for DR (t, k), with t de?ned in the region ?β ≤ Imt ≤ 0 > DR (t) =
[ θ(ε) + n(ωk ? iεsξ) ]e?iεwk t?ξst ?
1 2ωk β
Xη e?sωη t (9)
where s = sign[ Re(t) ], ξ is the regulator and Xη is the sum Xη = 8ξωη ωk rs = 2 + ξ 2 ][(iω + ω )2 + ξ 2 ] [(iωη ? ωk ) η k r,s=±1 iωη ? sωk + iξr (10)
where ωη = 2πη = 2πηT denotes the Matsubara frequencies. This term arises β from the residues of the distribution function when we integrate k0 in the complex plane. We will discuss its contribution later.
The bubble term
The rest of the paper will be the calculation of the bubble diagram, which is nothing else but the product of two propagators. In order to justify its relevance, we will brie?y mention where it comes from. We consider the two scalar ?eld theory described by the Lagrangian 1 1 1 L[φ, η] = ?? η? ? η ? m2 η 2 ? gφη 2 + L0 2 2 2 (11)
where L0 denotes the free Lagrangian for φ. If we integrate out the η-?eld ?uctuations and use a one-loop approximation we ?nd that the generating functional can be expressed as Z=
′ where Sef f is given by C
DφeiS0 [φ]+iSef f [φ]
i ′ Sef f [φ] = Trln[1 ? g?c (x, x′ )φ(x′ )] 2 7
and ?c (x, x′ ) is the propagator for the η ?eld. Expanding the logarithm we get
∞ ′ Sef f [φ] = p=1
where p denotes the number of the propagators. In this expression we will (2) concentrate on the ?rst non-local term which is the quadratic part Sef f of the expansion and is given by Sef f =
?ig 2 T r(?c (x, x′ )φ(x′ )?c (x′ , x)φ(x)) 4
One can permute the order of the elements inside the trace using the identity φ(x)?c (k) = ?c (k + p)φ(x) (16)
which is equivalent to the Taylor expansion of Fraser  for moving momentum operators to the left of functions depending on x, when we identify ′ p? = ?i?? . Thus the quadratic part of Sef f [φ] can be rewritten as Sef f = ?
d4 x′ φ(x)iB(p, β)φ(x′ )
with iB(p, β) being the bubble term given in terms of the propagators as iB(p, β) = g 2 dk0 2π d3 k ?c (k, m)?c (k + p, m) (2π)3 (18)
Separating the time dependence which interests us at ?nite temperature, Sef f is written Sef f = ? ×
ig 2 ti ?iβ d3 p dt φ(p, t) 4 ti (2π)3 d3 k ti ?iβ ′ dt ?c (k; t, t′ )?c (k + p; t′ , t)φ(p, t′ ) 3 ti (2π)
and the ?elds φ are periodic over the time path [ti , ti ? iβ]. This derivative expansion of the bubble term is well established at zero temperature. At ?nite temperature this is not so, since we will be expanding our theory around 8
an ill-de?ned point. This can be seen for example if we look at the ?11 component of the propagator of our theory which is given by ?(k, m) = k2 1 ? 2iπn(k0 )δ(k 2 ? m2 ) ? m2 + iε (20)
Substituting the propagator in Eq. (17), the temperature-dependent real part of the quadratic thermal e?ective action is given by the following expression which is nonanalytic at the zero four-momentum limit  Re(Sef f (2) [φ]) = ? with f (k) = where ω 2 = k2 + m2 and R=
2 (?0 ? ? 2 + 2isω?0 + 2irk(?? 2 )1/2 )r ? r,s=+1,?1
g2 32π 2
dkf (k)φ(x) φ(x)
kn(ω)Re(lnR) ω(?? 2 )1/2 ?
This result suggests that the derivative expansion breaks down at ?nite temperature, due to the problematic product of the two delta functions contained in the bubble. If the derivative expansion of the e?ective action is not rigorously possible, then the de?nition of an e?ective potential, the lowest order term in such an expansion, is not unique. This would have consequences in any kind of study concerning symmetry breaking and restoration, unless there is a formal way to overcome these pathologies and have a well de?ned derivative expansion.
E?ective potential term
We use Le Bellac and Mabilat’s formulation  to prove explicitly the ti independence of the e?ective potential in the case of two propagators and one external time t0 (?rst term of the bubble diagram) when adding both contributions from the horizontal and vertical path. Since we are interested
in time dependent ?elds, we will concentrate on the time integral of the bubble term in Eq. (19). This is given by GR = C
C c c DR (t0 ? t1 )DR (t0 ? t1 )dt1
where C will be in the region [ti , t0 ] and [t0 , ti ] for the horizontal path and [ti , ti ? iβ] for the vertical one as shown in Fig.1. Using the representation for the propagators of Eq. (9), the integrations over the two paths give GR H for the horizontal and GR for the vertical one  V GR = ? H AH 1 ? e?βω(ε1 +ε2 ) e2iξβ ( )[ 1 ? e?iω(ε1 +ε2 )(t0 ?ti ) e2ξ(ti ?t0 ) ] (2ω)2 iω(ε1 + ε2 ) + 2ξ AV eβω(ε1 +ε2 ) e?2iξβ ? 1 ?iω(ε1 +ε2 )(t0 ?ti ) 2ξ(ti ?t0 ) ( )[ e e ] (2ω)2 iω(ε1 + ε2 ) + 2ξ [ θ(±ε1 ) + n(ω ? iξε1 ) ][ θ(±ε2 ) + n(ω ? iξε2) ]
GR = ? V where
are ti -independent coe?cients. The KMS relation which reads as [ θ(ε) + n(ω ? iξε) ] = eεβω e?iξβ [ θ(?ε) + n(ω ? iξε) ] gives for the coe?cients AH and AV AH = e(ε1 +ε2 )βω e?2iξβ AV (29) (28)
Adding the ti -dependent terms of both paths Eq. (25) and Eq. (26) and using the KMS condition of Eq. (28), we ?nd that they cancel. Therefore, it is essential that we add the vertical contribution to ensure the ti -independence of our result. The remaining part of the sum is given by GR = GR + GR = ? H V 1 (AH ? AV ) 2 iω(ε + ε ) + 2ξ (2ω) 1 2 (30)
and using the de?nitions of AH and AV from Eq. (27) and identities of the θ functions, the sum is written as GR + GR = ? H V 1 (2ω)2 iβn(ω)(1 + n(ω)) (1 + 2n(ω))iω + 2iξ 2 n′ (ω) + (31) ?ω 2 ? ξ 2 2ω 2 10
In Eq. (31) the ?rst term is the result of Eq. (30) for ε1 = ε2 and the second term is the result for ε1 + ε2 = 0. Taking the limit of the regulator ξ to zero at the end, we have GR + GR = H V i iβn(ω)(1 + n(ω)) (1 + 2n(ω)) + 3 4ω 2ω 2 (32)
The previous result agrees completely with the ITF result for the e?ective potential , which proves the consistency of our theory to this order. Now we examine the cases of taking di?erent limits for the ti and the regulators and try to explain their physical meaning. 1. We take the limit ti → ?∞ (keeping the regulators ?nite) which should give us the real time formalism. Since the total sum is ti -independent, the limit of ti → ?∞ is already given by Eq. (31) in which the regulator can be taken to zero since there is no need for it any more after the limit has been performed. We notice from Eq. (26) that keeping the regulators ?nite the vertical part vanishes in this limit, recovering thus the RTF and the total sum is being given by the ti -independent contribution of the horizontal part. In the case of unregularised propagators, the vertical contribution contains a ti -independent term of the form β 1 n(ω)(1 + n(ω))2πδ(k 2 ? m2 ) i 2ω which in the regularised approach is hidden in the two horizontal parts of the contour, as seen in the last part of Eq. (31). We see that, to this order, the regularised formalism is dealing with the pathologies of the problematic delta functions, recovering RTF in the appropriate limit. 2. Now keeping ti ?nite, we take the zero limits of the regulators in different orders and ?nd GR (ε1 + ε2 = 0, ξ → 0) = GR (ξ → 0, ε1 + ε2 = 0) = 0 H H and the only contribution comes from the vertical part in the limit ε1 + ε2 = 0, ξ → 0 which is the second term of Eq. (31)
ξ→0,ε1 +ε2 =0
iβn(ω)(1 + n(ω)) 2ω 2
This limit is one part of the full e?ective potential of Eq. (32) as expected since it only corresponds to the limit of equal and opposite energies ω for the two propagators (ε1 + ε2 = 0).
The second derivative term
Now the same formalism will be used for the derivation of the second derivative term of the e?ective action in our 1-loop case, where only time-dependent ?elds are considered. Because now the sum will also contain terms polynomial in (ti ? t0 ) as well as exponential ones, the equivalence with the RTF and ITF is less straightforward. We will show that the RTF limit can be extracted in this case without problems of divergences as long as we keep the regulators ?nite and take them to zero after the limit has been done. The second derivative term will look like Γ2 =
(2) C c c dt1 DR (t0 , t1 )(t0 ? t1 )2 DR (t1 , t0 )
2 where we have omitted the 1 ?t φ(t0 ) factor of this term in the expansion. 2 (2) Using the de?nition of Eq. (1), Γ2 can be written as dk0 dk1 (2) Γ2 = dt1 (θc (t0 , t1 ) + n(k0 ))(θc (t1 , t0 ) + n(k1 )) (2π)2 C ×(t1 ? t0 )2 e?i(k0 ?k1 ?iε)(t0 ?t1 ) ρ(k0 )ρ(k1 ) (34)
where the regulator ε is used so that the limit of ti → ?∞ can be taken without problems. In the end it will be set to zero. The other two regulators ε1 and ε0 in the delta functions of ρ(k0 )ρ(k1 ) make sure that no problems appear in the equal energy (mass) case k0 = k1 = w. Now we ?rst perform the dt1 integration and then “absorb” the (t1 ? t0 )2 term by di?erentiating the result with respect to k0 . If we name the time integrals over the two paths as IC , we then write ?2 I (35) 2 C ?k0 C Substituting this formula into our general expression Eq. (34), the contributions from the di?erent paths can now be written dt1 (t1 ? t0 )2 θc (t1 , t0 )e?i(k0 ?k1 ?iε)(t0 ?t1 ) = i2 ΓH =
dk0 dk1 ?2 (n(k0 ) ? n(k1 ))ρ(k0 )ρ(k1 )i2 2 IH (2π)2 ?k0 12
dk0 dk1 ?2 [ n(k0 )(n(k1 ) + 1)]ρ(k0 )ρ(k1 )i2 2 IV (2π)2 ?k0
In ΓH the θ2 term vanishes due to the opposite sign of its time arguments. The n2 term vanishes due to the cancellation between the two horizontal (2) paths. In ΓV one of the θ-functions always vanishes due to the choice of t0 on the horizontal path. The time integrations IH , IV over the two paths give IH = i (1 ? e?i(k0 ?k1 ?iε)(t0 ?ti ) ) k0 ? k1 ? iε
(1 ? eβ(k0 ?k1 ?iε) ) k0 ? k1 ? iε The analytical calculation of the di?erent path contributions is quite complicated since it involves ?rst and second order residues and therefore derivatives of the distribution functions. We performed the momentum integrations and then took the same limits of our variable ?t = ti ?t0 and of the regulators as before to check the consistency of our method for the second derivative term. IV = ie?i(k0 ?k1 ?iε)(t0 ?ti ) × 1. We took the ?t → ?∞ limit keeping the regulators ?nite. In the total sum the ?t-dependence appears in terms like ?tn eε?t and ?tn (n = 0, 1, 2). These terms could cause divergences in the ?t → ?∞ limit but they disappear once we include the vertical part in our calculation. Our result is independent of the order in which the regulators are taken to zero in the end and is given by a ?nite term coming from the horizontal part i (1 + 2n(ω)) lim Γ(2) = ? ti →?∞ 8 ω5 This term looks like the ?rst order term of the e?ective potential divided by ω 2, as it can be seen from Eq. (32), which is sensible since it is essentially the ?rst correction due to the second derivative. 2. Our second limit is ti = t0 , ε = 0 in order to try to recover the ITF result (ti = t0 = ?nite and ε is not needed any more since ti is ?nite).
This proved to be also independent of the order of the zero limits of the regulators. We obtained Γ(2) (ti = t0 , ε = 0; ε1 = 0; ε0 = 0) = i ? 5 [ (2n(ω) + 1) ? βω(2n(ω)(n(ω) + 1) + 1) 8ω 4β 3 ω 3 n(ω)(n(ω) + 1) + β 2 ω 2 (2n(ω) + 1) + ] 3 which is consistent with the derivation of the second derivative term in the ITF formalism .
Another possible way of performing our calculation is to consider the full Taylor series of the ?eld but do the actual expansion and study the individual terms in the end. We will generalise our method considering di?erent energies (ω and ?) in the delta functions of Eq. (3) for each propagator of the bubble term. In this way we will check the analyticity limits of the full derivative term by taking the limits ? → ±ω (? → 0) and ?t → 0 in di?erent orders ? at the end of the calculation. The expanded ?eld can be written as
φ(t1 ) =
1 ?n = e(t1 ?t0 )?t φ(t) (t1 ? t0 )n n φ(t) t=t0 n! ?t t=t0
The full derivative term of the ?eld inserted between the two propagators, which is the last time integral in Eq. (19), looks like Γ(B) =
C c dt1 DR (t0 , t1 ) e(t1 ?t0 )?t φ(t) t=t0 c DR (t1 , t0 )
This term acts as an energy-shift by ?i?t in the exponentials of the propagators making the time-integrals over the paths IC of Eq. (35) look like
C ′ dt1 θc (t1 , t0 )e?i(k0 ?k1 ?i(ε+?t ))(t0 ?t1 ) = IC
Performing the time integration for the horizontal and vertical path as before, we get (1 ? e?i(k0 ?k1 ?i(ε+?t ))(t0 ?ti ) ) ′ IH = i k0 ? k1 ? i(ε + ?t ) and (1 ? eβ(k0 ?k1 ?i(ε+?t )) ) ′ IV = ie?i(k0 ?k1 ?i(ε+?t ))(t0 ?ti ) × k0 ? k1 ? i(ε + ?t ) The energy integration gives us the full bubble term as a sum over the two paths written in terms of ?t = ti ? t0 Γ(B) = ΓH + ΓV with ΓH = and ΓV where A = ω + ? ? i(ε1 + ε0 ? ε ? ?t ) Now we can check the analyticity of our result keeping ?t ?nite. We expand the distribution functions and take the limits of our regulators to zero (we can do that since we keep ?t ?nite). If we take the limits ? → ±ω and ?t → 0, we get ?nite and independent of the order of the limits results. The full derivative expansion of the bubble term, therefore, is analytical in this limit and is ΓH (? → ±ω, ?t → 0) = i for the horizontal case and ΓV (? → ±ω, ?t → 0) = i for the vertical one. 15
(B) (B) (B) (B) (B) (B)
in(ω ? iε0 )n(? ? iε1 ) (eβ(ω+??i(ε0 +ε1 )) ? 1)(1 ? e?iA?t ) ×[ ] (40) 4ω? A ±ω,?
in(w ? iε0 )n(? ? iε1 ) e?iA?t e?iβ(ε+?t ) (eβA ? 1) ×[ ] 4ω? A ±ω,?
(2n(ω) + 1)(1 ? cos(2ω?t)) 2ω 3
(2n(ω) + 1)cos(2ω?t) βn(ω)(n(ω) + 1) +i (43) 3 2ω w2
1. Now we consider the limit ti = t0 in Eq. (42) and Eq. (43). This gives ΓH = 0 and
βn(ω)(1 + n(ω)) (2n(ω) + 1) +i (45) 3 2ω w2 This is exactly the result for the e?ective potential using the ITF formalism, as expected since it is the zeroth time and space-derivative term, when ti = t0 . It also agrees with our previous result in Eq. (32) of the e?ective potential after we set the regulators to zero. (In Eq. (32) the result di?ers by a factor of 1/2 due to the fact that we have initially considered same energies ω for the propagators and this corresponds to half of the result of the e?ective potential of Eq. (45) when di?erent energies are assumed). ΓV =i 2. Now we take the limit ?t → ?∞ and we will check the analyticity (B) again. In this case only the ?rst ?t-independent part of ΓH survives and taking the regulators to zero after the limit has been performed, we get i (n(ω) ? n(??)) Γ(B) (?t → ?∞) = (46) ±ω,? 4ω? (ω + ? + i?t ) The analyticity check for Eq. (46) gives ?nite but di?erent results for di?erent orders of performing the limits (? → ±ω, ?t → 0). We found that performing the time-derivative (?t → 0) limit ?rst and the spatialderivative (? → ±ω) afterwards, we had the usual e?ective potential term of Eq. (45) lim Γ(B) = i (2n(ω) + 1) βn(ω)(1 + n(ω)) +i 3 2ω w2
but reversing the order of the limits gave us only the ?rst term of our previous result (2n(ω) + 1) lim Γ(B) = i ?→±ω,?t →0 2ω 3 We see that although we don’t have divergence problems in taking the limits in both orders, approaching the zero from the space-derivative 16
?rst seems to produce only part of the full result in agreement with the result of Evans using ITF . Now we take only the spatial derivative to zero in the ?t → ?∞ case of Eq. (46) which gives lim Γ(B) = 2i (2n(ω) + 1) 2 ω(4ω 2 + ?t )
If we now expand our result in powers of the time-derivative ?t , we get the zeroth order term of our analyticity check and a second order term of the form i(2n(ω) + 1) 2 ? (?t ) 8ω 5 This is exactly the second order time-derivative term derived in this limit using our previous method in section 2.4. We have to note that in the case of ?t → ?∞, there is no term linear in ?t . Now we perform the same expansion in powers of the time-derivative up to the second order but for a general ?nite ?t, for both horizontal and vertical paths in Eq. (40) and Eq. (41) and take the limits (? → ±ω) to get the e?ective potential and the higher derivative terms. We identify the terms as follows
0 1. ?t term
Γ0 = ?i H Γ0 = i V
1 2. ?t term
(2n(ω) + 1) 2iω?t (e + e?2iω?t ? 2) 3 4ω
βn(ω)(n(ω) + 1) (2n(ω) + 1) 2iω?t (e + e?2iω?t ) + i 3 4ω ω2
Γ1 = H
(2n(ω) + 1) 2iω?t [(e ? e?2iω?t ) ? 2iω(?t)(e2iω?t + e?2iω?t )] (49) 8ω 4
1 [(2n(ω) + 1)(e2iω?t ? e?2iω?t ) 8ω 4 ? 2βω((n(ω) + 1)2 e2iω?t ? n2 (ω)e?2iω?t ) ? 4β 2 ω 2 n(ω)(n(ω) + 1)] i [(2n(ω) + 1)(e2iω?t + e?2iω?t ) + 3 4ω + 4βωn(ω)(n(ω) + 1)](?t) (50) = ?
We notice the existence of a non-zero ?t -dependent term unlike the zero temperature case where such a term vanishes. This could be related to the loss of Lorentz invariance in the ?nite temperature case and could be interpreted as an energy shift. The existence of such a linear term might be of great physical importance in the study of time-dependent systems. Such a term did not exist in the expansion for the ?t in?nite case, where only zero and second order terms in the time-derivative survived. This makes sense since in the in?nite time limit any interaction with the heat bath which gives rise to such linear terms will have been damped. Mathematically this term could arise due to the shape of the time contour, which in the ?nite ?t case is non-symmetric. However this is not the case for the zero-temperature situation or the non zero temperature one in the in?nite ?t limit where the symmetry of the contour will make any time integration of odd terms in the derivative expansion to vanish. In the ?t = 0 case this term is equal to ?iβΓ0 2 0 where Γ is the e?ective potential term given by Eq. (47) and Eq. (48) in the ?t = 0 limit. Γ1 =
2 3. ?t term
Γ2 = H
i(2n(ω) + 1) 2iω?t [(e + e?2iω?t ? 2) ? 2iω(?t)(e2iω?t ? e?2iω?t ) 5 16ω ? 2ω 2(?t)2 (e2iω?t + ε?2iω?t )] (51) = Γ2 1 + Γ2 2 V V 18 (52)
with Γ2 1 = ? V i [(2n(ω) + 1)(e2iω?t + e?2iω?t ) ? 2βω((n(ω) + 1)2 e2iω?t 16ω 5 + n2 (ω)e?2iω?t ) + 2β 2 ω 2((n(ω) + 1)2 e2iω?t ? n2 (ω)e?2iω?t ) 8 3 3 + β ω n(ω)(n(ω) + 1)] 3 and Γ2 2 = ? V 1 [(2n(ω) + 1)(e2iω?t ? e?2iω?t ) ? 2βω((n(ω) + 1)2 e2iω?t 4 8ω ? n2 (ω)e?2iω?t ) ? 4β 2 ω 2 n(ω)(n(ω) + 1)](?t) i [(2n(ω) + 1)(e2iω?t + e?2iω?t ) + 8ω 3 + 4βωn(ω)(n(ω) + 1)](?t)2 (54) (53)
If we take the ti = t0 limit of our second derivative term, we recover our previous derivation of the same term in section 2.4. In our calculation we have omitted the contribution of the Xη term of Eq. (10). This term which arises from the residue of the distribution function vanishes since it is proportional to the regulator. In the ?nite ?t case the regulators are set to zero before any limit is taken while in the in?nite ?t case they are set to zero once the in?nity limit has been performed. In both cases this term does not contribute.
The initial time dependence
In this section we will treat the initial time-dependence of our problem in a rather more formal way. Based on Le Bellac and Mabilat’s proof of the ti -independence of a regularised Green function , we will prove the same for our e?ective potential term. Our ti -dependent integrals in this case are
dt1 GR (t1 , t0 )GR (t1 , t0 ) + 19
ti ?iβ t0
dt1 GR (t1 , t0 )GR (t1 , t0 )
Di?erentiating the ?rst term with respect to ti we get ?G< (ti , t0 )G< (ti , t0 ) R R Repeating for the second term we now get G> (ti ? iβ, t0 )G> (ti ? iβ, t0 ) R R Using the KMS condition for thermal equilibrium G> (t ? iβ) = G< (t) R R we see that these terms cancel. If we repeat the same method for the higher derivative terms of the ?eld of the bubble case, we get a non-zero result which shows explicitly the ti -dependence of these terms. The same analysis for the second derivative term gives G< (ti , t0 )[?β 2 ? 2iβ(ti ? t0 )]G< (ti , t0 ) R R (56)
If we generalise in the case of the m-th derivative term, the ti dependence of the derivative with respect to ti will have the form
m k?1 < GR (ti , t0 )[ k=1 j=0
(m ? j) (?iβ)k (ti ? t0 )m?k ]G< (ti , t0 ) R k!
We see that the individual terms of the expanded ?eld are clearly ti -dependent even in the case of ti = t0 , where the highest order β-term survives in the previous sum. This is somehow expected since a truncated expansion of the ?eld makes it no longer periodic. On the other hand if we have a periodic ?eld φ(t) in equilibrium, the derivative with respect to ti discussed earlier will give ? G< (ti , t0 )φ(ti )G< (ti , t0 ) + G> (ti ? iβ, t0 )φ(ti ? iβ)G> (ti ? iβ, t0 ) R R R R (58)
which is zero for periodic ?elds and regularised propagators obeying the KMS condition.
Conclusions and possible applications
We found that using this closed time path formalism we can avoid the pathologies in the RTF and derive the ITF limit as well, both in the case of the e?ective potential and in the second derivative correction of the bubble diagram. The fact that we can cancel the divergence in our e?ective action using our prescription, hence providing us with a formalism which allows us to compute quantum corrections to the e?ective potential is the key point of our paper. However we found that the inclusion of the vertical path and the careful treatment of the regulators are essential for the cancellations to happen. We showed a general way to compute higher derivative terms in the bubble and derived the complete bubble term. We checked its analyticity for ?nite and in?nite time di?erences ?t and found di?erent limits in the second case. The non zero, linearly dependent on the time derivative of the ?eld term found in the ?nite ?t case, may be related to the loss of translational invariance at ?nite temperature. The physical meaning of such a term and in particular its sign and whether it is complex or real may be important in the study of time-dependent systems. Gribosky and Holstein  do not ?nd such a linear term in their expansion of derivatives of the ?eld. Motivated by the use of Feynman parametrisation at zero temperature  they calculate the vacuum polarisation diagram using ITF but extending to continuous p0 ?rst and evaluating the mode sum afterwards. They compare their result with Dittrich’s  background ?eld method of calculating the e?ective Langrangian at ?nite temperature and in both cases there is no linear term unlike our case which appears for any ?nite ?t. The extension of our calculation to higher derivative terms and to spacedependent ?elds will give a full e?ective action whose importance in ?eld theory was discussed earlier. Our calculation may be performed for higher order diagrams in the expansion of the one-loop e?ective action, but this is beyond the scope of this paper. We have considered a two real scalar ?eld theory, but we could in principle use our method in di?erent models, such as a Yukawa or a gauge theory or even consider systems with time-dependent parameters. The possibility of evaluating quantum corrections can be more directly applied to phase transitions, where they may indicate us something 21
about the order of the transition.
We would like to thank T. S. Evans and R. J. Rivers for many helpful discussions, A. Gomez Nicola for his interesting remarks and B. D. Wandelt for his help with Mathematica. This work has also been supported in part by the European Commission under the Human Capital and Mobility programme, contract number CHRX-CT94-0423.
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